case out Kummer

Project.toml

@@ -1,6 +1,6 @@
 name = "HypergeometricFunctions"
 uuid = "34004b35-14d8-5ef3-9330-4cdb6864b03a"
-version = "0.3.6"
+version = "0.3.7"
 
 [deps]
 DualNumbers = "fa6b7ba4-c1ee-5f82-b5fc-ecf0adba8f74"

README.md

@@ -4,4 +4,4 @@
 
 A Julia package for calculating hypergeometric functions
 
-This package implements the generalized hypergeometric function `pFq([a1,…,am], [b1,…,bn], z)`. In particular, the Gauss hypergeometric function is available as `_₂F₁(a, b, c, z)`, and also `_₃F₂([a1, a2, a3], [b1, b2], z)`.
+This package implements the generalized hypergeometric function `pFq([a1,…,am], [b1,…,bn], z)`. In particular, the Gauss hypergeometric function is available as `_₂F₁(a, b, c, z)`, Kummer's confluent hypergeometric function is available as `_₁F₁(a, b, z)` and `HypergeometricFunctions.M(a, b, z)`, as well as `_₃F₂([a1, a2, a3], [b1, b2], z)`.

src/HypergeometricFunctions.jl

@@ -7,10 +7,11 @@ module HypergeometricFunctions
 
 using DualNumbers, LinearAlgebra, SpecialFunctions
 
-export _₂F₁, _₃F₂, pFq
+export _₁F₁, _₂F₁, _₃F₂, pFq
 
 include("specialfunctions.jl")
 include("gauss.jl")
+include("kummer.jl")
 include("generalized.jl")
 include("drummond.jl")
 include("weniger.jl")

src/generalized.jl

@@ -10,6 +10,8 @@ function pFq(α::AbstractVector, β::AbstractVector, z; kwds...)
         return exp(z)
     elseif length(α) == 1 && length(β) == 0
         return exp(-α[1]*log1p(-z))
+    elseif length(α) == 1 && length(β) == 1
+        return _₁F₁(α[1], β[1], float(z))
     elseif length(α) == 2 && length(β) == 1
         return _₂F₁(α[1], α[2], β[1], float(z))
     elseif abs(z) ≤ ρ

src/kummer.jl

@@ -0,0 +1,12 @@
+"""
+Compute Kummer's confluent hypergeometric function `₁F₁(a; b; z)`.
+"""
+function _₁F₁(a, b, z)
+    if real(z) ≥ 0
+        return _₁F₁maclaurin(a, b, z)
+    else
+        return exp(z)*_₁F₁(b-a, b, -z)
+    end
+end
+
+const M = _₁F₁

src/specialfunctions.jl

@@ -147,7 +147,7 @@ function unsafe_gamma(x::BigFloat)
     return z
 end
 unsafe_gamma(z::Dual) = (r = realpart(z);w = unsafe_gamma(r); dual(w, w*digamma(r)*dualpart(z)))
-unsafe_gamma(z) = gamma(z) 
+unsafe_gamma(z) = gamma(z)
 """
     @lanczosratio(z, ϵ, c₀, c...)
 
@@ -492,6 +492,17 @@ function _₂F₁taylor(a::Number, b::Number, c::Number, z::Number)
     return S₁
 end
 
+function _₁F₁maclaurin(a::Number, b::Number, z::Number)
+    T = float(promote_type(typeof(a), typeof(b), typeof(z)))
+    S₀, S₁, j = one(T), one(T)+a*z/b, 1
+    while errcheck(S₀, S₁, 10eps(real(T))) || j ≤ 1
+        rⱼ = (a+j)/((b+j)*(j+1))
+        S₀, S₁ = S₁, S₁+(S₁-S₀)*rⱼ*z
+        j += 1
+    end
+    return S₁
+end
+
 function _₃F₂maclaurin(a₁, a₂, a₃, b₁, b₂, z)
     T = float(promote_type(typeof(a₁), typeof(a₂), typeof(a₃), typeof(b₁), typeof(b₂), typeof(z)))
     S₀, S₁, j = one(T), one(T)+(a₁*a₂*a₃*z)/(b₁*b₂), 1